amongst random length-k (k>2) sequences of independent coin tosses with at least one heads before toss k, the expected proportion of (heads after heads)/(tosses after heads) is less than 1⁄2.
Does this need to be k>3? Checking this for k=3 yields 6 sequences in which there is at least one head before toss 3. In these sequences there are 4 heads-after-heads out of 8 tosses-after-heads, which is exactly 1⁄2.
Edit: Ah, I see this is more like a game score than a proportion. Two “scores” of 1 and one “score” of 1⁄2 out of the 6 equally likely conditional sequences.
Does this need to be k>3? Checking this for k=3 yields 6 sequences in which there is at least one head before toss 3. In these sequences there are 4 heads-after-heads out of 8 tosses-after-heads, which is exactly 1⁄2.
Edit: Ah, I see this is more like a game score than a proportion. Two “scores” of 1 and one “score” of 1⁄2 out of the 6 equally likely conditional sequences.